Aniceto Murillo

Lie Models for the Components of Sections of a Nilpotent Fibration

We give an explicit Lie model for any component of the space of free and pointed sections of a nilpotent fibration, and in particular, of the free and pointed mapping spaces. Among the applications presented, we obtain a Lie model of the exponential law and prove that, in many cases, the rank of the homotopy groups of the mapping space grows at the same rate as the rank of the homotopy groups of the target space.

A bound for the nilpotency of a group of self homotopy equivalences

Let E-Omega(X) be the group of homotopy classes of self-homotopy equivalences of X such that Omega f similar or equal to 1d(Omega X). We prove that epsilon(Omega)(X) is a nilpotent group and that nil epsilon(Omega)(X) less than or equal to cat(X) - 1.

On the evaluation map

The evaluation map of a differential graded algebra or of a space is described under two different approaches. This concept turns out to have geo- metric implications: (i) A 1-connected topological space, with finite-dimensional rational homotopy, has finite-dimensional rational cohomology if and only if it has nontrivial evaluation map. (ii) Let E -^ B be a fibration of simply- connected spaces. If the rational cohomology of the fibre is finite dimensional and the evaluation map of the base is different from zero, then the evaluation map of the total space is nonzero. Also, if p is surjective in rational homotopy and the evaluation map of E is nontrivial, then the evaluation map of the fibre is different from zero.

Complexity in rational homotopy

Abstract The computation of classical invariants of the rational homotopy type of simply connected spaces is shown to be an NP-hard problem.

Maurer-Cartan elements in the Lie models of finite simplicial complexes

In a previous work, we have associated a complete differential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we have also a realization functor from the category of complete differential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex.

Gorenstein graded algebras and the evaluation map

We consider graded connected Gorenstein algebras with respect to the evaluation map ev(G) = Ext(G)(k,epsilon) :: Ext(G)(k,G) --> Ext(G)(k,k). We prove that if ev(G) not equal 0, then the global dimension of G is finite.

Rational fibrations in differential homological algebra

In this paper, a result of [6] is generalized as follows: Given a fibration F → E → ρ → B of simply connected spaces in which either, the fibre has finite dimensional rational cohomology, or, it has finite dimensional rational homotopy and ρ induces a surjection in rational homotopy, we construct an explicit isomorphism, φ:Ext C * (B,Q) (Q,C*(B;Q))⊗#6AExt C * (F;Q) (Q,C*(F,Q)) → ≅ Ext C*(E;Q) (Q,C*(E;Q). This is deduced from its «algebraic translation», a more general result in the environment of graded differential homological algebra

The virtual Spivak fiber, duality on fibrations and Gorenstein spaces

In this paper we study a generalization of the homology of the Spivak fiber of a 1-connected space over any field and deduce consequences concerning Poincare complexes, Gorenstein spaces and finiteness properties on fibrations.

The evaluation map of some Gorenstein algebras

Abstract We define the Ext-Milnor–Moore spectral sequence and prove that, over anyfield K , a differential graded algebra, with a finite-dimensional Lie algebra, is Gorenstein, and it has finite-dimensional cohomology if and only if it has non-zero evaluation map.

The Hilali Conjecture for Hyperelliptic Spaces

The Hilali conjecture predicts that for a simply connected elliptic space, the total dimension of the rational homotopy does not exceed that of the rational homology. Here, we give a proof of this conjecture for a class of elliptic spaces known as hyperelliptic.